Introduction to CLASSICAL MECHANICS

This book is, in essence, an updated and revised version of an earlier textbook, Newtonian Mechanics, written about fifteen years ago by one of us (APF) and published in 1971. The book has been significantly changed in emphasis as well as length. Our aim has been to produce a mechanics text, suitable for use at beginning university level, for students who have a background typified by the British sixth-form level in physics and mathematics. We hope, however, that the book will also be found useful in the teaching of mechanics at the upper levels of the secondary schools themselves. Calculus is freely used from the outset. In making the present revision we have drastically cut down on the amount of historical and more discursive material. Nevertheless, our goal has been to present classical mechanics as physics, not as applied mathematics. Although we begin at the beginning, we have aimed at developing the basic principles and their applications as rapidly as seemed reasonable, so thatby the end of the book students will be able to feel that they have achieved a good working knowledge of the subject and can tackle fairly sophisticated problems. To help with this process, each chapter is followed by a good number of exercises, some of them fairly challenging. We shall be very grateful to receive comments and corrections from those who use this book.

1 Space, time and motion.- What is motion?.- Frames of reference.- Coordinate systems.- Combination of vector displacements.- Scalar product of vectors.- Units and standards of length and time.- Velocity.- Relative velocity and relative motion.- Acceleration.- Straight-line motion.- Uniform circular motion.- Velocity and acceleration in polar coordinates.- Problems.- 2 Forces.- Forces in equilibrium.- Action and reaction in the contact of objects.- Rotational equilbrium: torque.- Inertia.- Force and inertial mass: Newton’s second law.- Some comments on Newton’s second law.- The invariance of Newton’s second law; relativity.- Concluding remarks.- Problems.- 3 Using Newton’s laws.- Some examples of F = ma.- Circular paths of charged particles in uniform magnetic fields.- The fracture of rapidly rotating objects.- Motion against resistive forces.- Detailed analysis of resisted motion.- Motion governed by viscosity.- Growth and decay of resisted motion.- Simple harmonic motion.- Problems.- 4 Universal gravitation.- The discovery of universal gravitation.- Kepler’s third law.- The moon and the apple.- The gravitational attraction of a large sphere.- Other satellites of the earth.- The value of G, and the mass of the earth.- Local variations of g.- Inertial and gravitational mass.- Weight.- Weightlessness.- The discovery of Neptune.- Gravitation outside the solar system.- Einstein’s theory of gravitation.- Problems.- 5 Collisions and conservation laws.- The conservation of linear momentum.- Action, reaction, and impulse.- Extending the principle of momentum conservation.- Jet propulsion.- Rockets.- The zero-momentum frame.- Kinetic energy of a two-body system.- Kinetic energy changes in collisions.- Interacting particles subject to external forces.- The neutrino.- Problems.- 6 Energy conservation in dynamics; vibrational motions.- Work, energy, and power.- Energy conservation in one dimension.- The energy method for one-dimensional motions.- Some examples of the energy method.- The harmonic oscillator by the energy method.- Small oscillations in general.- The linear oscillator as a two-body problem.- Problems.- 7 Conservative forces and motion in space.- Extending the concept of conservative forces.- Object moving in a vertical circle.- The simple pendulum.- The pendulum as a harmonic oscillator.- The simple pendulum with larger amplitude of swing.- Universal gravitation: a conservative central force.- A gravitating spherical shell.- A gravitating sphere.- Escape speeds.- More about the criteria for conservative forces.- Fields.- Motion in conservative fields.- The effect of dissipative forces.- Problems.- 8 Inertial forces and non-inertial frames.- Motion observed from unaccelerated frames.- Motion observed from an accelerated frame.- Accelerated frames and inertial forces.- Accelerating frames and gravity.- Centrifugal force.- General equation of motion in a rotating frame.- The earth as a rotating reference frame.- The tides.- Tidal heights; effect of the sun.- The search for a fundamental inertial frame.- Problems.- 9 Motion under central forces.- Basic features of the problem.- The conservation of angular momentum.- Energy conservation in central force motions.- Use of the effective potential-energy curves.- Bounded orbits.- Unbounded orbits.- Circular orbits in an inverse-square force field.- Elliptic orbits: analytical treatment.- Energy in an elliptic orbit.- Possible orbits under a 1/r2 force.- Rutherford scattering.- Problems.- 10 Extended systems and rotational dynamics.- Momentum and kinetic energy of a many-particle system.- Angular momentum.- Angular momentum as a fundamental quantity.- Conservation of angular momentum.- Moments of inertia of extended objects.- Two theorems concerning moments of inertia.- Kinetic energy of rotating objects.- Angular momentum conservation and kinetic energy.- Torsional oscillations and rigid pendulums.- Linear and rotational motions combined.- Background to gyroscopic motion.- Gyroscope in steady precession.- Atoms and nuclei as gyroscopes.- The precession of the equinoxes.- Problems.- Solutions to problems.